Groups in which all involutions are 3-transvections

Let G be a group which is generated by the set of its involutions, and assume that the set of integers which occur as orders of products of two involutions in G is \(\{1,2,3,4\}\) $\{1,2,3,4\}$ . It is shown that \(G\simeq \textrm{PSL}(2,7)\) $G\simeq \text{PSL}(2,7)$ or \(G\simeq \textrm{PSU}(3,3)\) $G\simeq \text{PSU}(3,3)$ or G is a \(\{2,3\}\) $\{2,3\}$ -group and \(G/O_{2}(G) \simeq S_{3}\) $G/O_2(G)\simeq S_3$ .